An entire infinite plane fit inside a circle — Escher's favorite geometry
The whole infinite hyperbolic plane fits inside this circle. The edge is infinitely far away, and straight lines show up as arcs.
Click pairs of points to draw geodesics. Draw several through one point and watch them fan out, then try some near the edge.
The Poincare disk model maps the entire infinite hyperbolic plane into the interior of a unit circle. Distances grow without bound as you approach the boundary, and straight lines (geodesics) appear as circular arcs that meet the boundary at right angles. This conformal model preserves angles, making it ideal for understanding hyperbolic geometry visually — and it was the model that inspired M.C. Escher's famous Circle Limit woodcuts.
Conformal model — Angles are preserved, so perpendicularity and angle measurement work just like in Euclidean geometry
In the Poincare disk, geodesics are circular arcs perpendicular to the boundary circle. Click pairs of points inside the disk to draw geodesics. Notice how arcs that look “bent” in Euclidean terms are actually the shortest paths in hyperbolic space.
Draw several geodesics through the same point and observe how they fan out. Try drawing geodesics near the boundary — they become nearly tangent to the circle, reflecting the infinite distance to the edge.
Geodesics — Shortest paths in the Poincare disk are circular arcs perpendicular to the boundary, or diameters through the center
Drag two points and compare their Euclidean and hyperbolic distances. Near the center they agree; near the edge, a tiny step becomes an enormous one.
Drag point A or B inside the disk. Concentric contours show lines of constant hyperbolic distance from A. The dashed line shows the Euclidean straight path; the solid arc is the hyperbolic geodesic.
Drag two points inside the disk to compare Euclidean distance |p - q| with hyperbolic distance d(p, q). Near the center, the two metrics nearly agree. Near the boundary, a tiny Euclidean step corresponds to an enormous hyperbolic distance — the metric blows up as 2/(1 - r²).
Hyperbolic distance — Grows logarithmically near the boundary; a Euclidean inch at the edge represents infinite hyperbolic distance
Pick a tool to draw geodesics, circles, midpoints, and perpendicular bisectors. Notice that the bisector is an arc, not a straight line.
Classical Euclidean constructions have hyperbolic counterparts. Use this tool to perform geodesic constructions: draw geodesics, construct hyperbolic circles, find geodesic midpoints, and draw perpendicular bisectors — all computed exactly in the Poincare disk model.
Key insight: The perpendicular bisector of a hyperbolic segment is the set of all points equidistant from both endpoints — just as in Euclidean geometry. But the bisector itself is a geodesic arc, not a straight line.
Change the tiling, motif, and palette, and turn on rotation. Every motif is the same hyperbolic size, even as they shrink toward the edge.
Escher Circle Limit — A GPU-rendered {p,q} hyperbolic tiling. The motifs are congruent in hyperbolic geometry but shrink in the Euclidean picture as they approach the boundary circle. Change p and q to explore different tilings (valid when 1/p + 1/q < 1/2).
M.C. Escher's Circle Limit series rendered a hyperbolic tiling with repeating motifs that shrink toward the boundary — each motif the same hyperbolic size, but appearing smaller in the Euclidean picture. This GPU-accelerated shader lets you explore {p, q} tilings with different motifs, color palettes, and automatic Mobius rotation.
Escher tilings — Regular {p, q} tilings tile the hyperbolic plane when 1/p + 1/q < 1/2, producing infinitely many congruent copies
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